The topology of reputation effects

Abstract

I study the topology of reputation effects in the canonical framework in which a long-lived player -- either a normal type who acts strategically or a commitment type who plays a fixed distribution over actions -- faces a sequence of short-lived players who may misspecify the commitment-type signal process. I show that reputation effects are robust to such misspecification in the entropy-rate topology on commitment-type signal processes; in contrast, they are discontinuous in finite-dimensional topology, equivalently weak convergence and, for compatible bounded metrics, Wasserstein and Prokhorov convergence: there exists a convergent sequence of subjective commitment processes along which the patient normal type's highest equilibrium payoff is at most his highest complete-information equilibrium payoff. Reputation effects are therefore an infinite-horizon statistical test.